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(1)On the Monopole Map in Three Dimensions. Dissertation zur Erlangung des Doktorgrades der Fakult¨at f¨ur Mathematik der Universit¨at Bielefeld. vorgelegt von Zvonimir Sviben. Betreuer: Prof. Dr. Stefan A. Bauer. Bielefeld, Dezember 2017.

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(3) Acknowledgements There are many people without whose help and support this thesis would have been merely a distant dream. First and foremost, I am deeply grateful to my supervisor Prof. Dr. Stefan A. Bauer for introducing me to the astonishing world of mathematical gauge theory, for sharing his marvellous mathematical insights, and for his unwavering enthusiasm and patience. I would also like to thank all the members of the Geometry and Topology group in Bielefeld for many instructive seminars and discussions as well as countless pleasant conversations. Special thanks goes to Hanno von Bodecker, Andriy Haydys, Stefan Behrens and Tyrone Cutler for their support, academic and otherwise. I am also grateful to Prof. Dr. Moritz Kaßmann for his continuous encouragement. The working environment at the University, as well as my spare time, were greatly enriched by my dear colleagues and friends Laura and Stephan, Christian, Bartek, Dainius, Chris, C´edric, Darragh, and Vanja. I am especially grateful to Matej and Lennart for highly interesting and entertaining chess games, inspiring discussions about chess history and many amusing anecdotes from tournament play. I wish to thank my neighbours Gerlinde, Irma, Dieter and Frank for making me feel like a part of their families. Aside from patiently helping me to improve my knowledge of the German language, their affection and thoughtfulness was invaluable in making me feel at home in Bielefeld. I am greatly indebted to my dearest and closest friends Antonijo, Danijela, Dubravka, Marko, and Marina. I cannot imagine finishing my studies without their unyielding encouragement, invaluable advice and the wonderful times we had. I should particularly like to thank my cousin Vladimir for being an inexhaustible source of optimism and energy. A big thank you also goes to my grandmother baka Maca for all her help throughout the years. Finally, I would like to express my gratitude to my brothers Branimir and ˇ Tomislav and my parents Dunja and Cedomir for their constant support and belief in me. The importance of their role during the preparation of this thesis, and in my life in general, cannot be adequately put into words..

(4) Contents. 0 Introduction 0.1 Historical background and motivation . . . . . . . . . . . . . . . . 0.2 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.3 A remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Preliminaries 1.1 Notation and remarks . . . . . . . . . . . . . . 1.2 Some identifications and conventions . . . . . 1.3 Scalar products and norms . . . . . . . . . . . 1.4 Clifford algebras and spin groups . . . . . . . . 1.4.1 Spin representation . . . . . . . . . . . 1.4.2 Other representations of the spin group 1.4.3 The associated bundles . . . . . . . . . 1.5 A different Clifford module structure on forms 1.6 Dirac operator on forms . . . . . . . . . . . . . 1.7 The quadratic term . . . . . . . . . . . . . . . . 1.7.1 Two ways of writing the quadratic term 1.7.2 Derivation of the quadratic term . . . . 1.7.3 Norm of the quadratic term . . . . . . .. 1 1 2 4. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. 5 5 6 7 9 14 17 17 19 20 21 22 23 26. 2 The monopole map on 3-manifolds 2.1 Assumptions and general context . . . . . . . . . . . . . . . 2.2 Seiberg-Witten equations on 3-manifolds . . . . . . . . . . 2.3 Monopole map for closed 3-manifolds . . . . . . . . . . . . 2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The refined Seiberg-Witten invariant for closed 3-manifolds. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 29 29 29 30 30 38 46. ii. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . ..

(5) Contents 3 New version of the monopole map 3.1 Definition, assumptions and some notation . . . 3.2 Boundedness property . . . . . . . . . . . . . . . 3.2.1 Adaptation of the boundedness property 3.2.2 A priori estimate . . . . . . . . . . . . . . 3.2.3 Bootstrapping . . . . . . . . . . . . . . . 3.3 Statement of the main result . . . . . . . . . . . 3.4 Renormalisation of the monopole map . . . . . .. iii. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 47 47 49 49 50 56 60 60. 4 The monopole map on a 3-torus 63 4.1 Notation and setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 The Dirac operators . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 The monopole map . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Appendix A A.1 Sobolev spaces and elliptic operators . . . . . . . . . A.1.1 Definition of Sobolev norms . . . . . . . . . . A.1.2 Stronger version of the elliptic estimate . . . . A.1.3 An equivalent definition of Sobolev norms . . A.1.4 Sobolev theorems . . . . . . . . . . . . . . . . A.2 Some facts from Hodge theory . . . . . . . . . . . . . A.3 Some relations between Clifford and exterior algebras A.4 The connection Laplacian . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 67 67 67 68 69 70 71 72 74. Endnotes. 75. Bibliography. 83. Notation. 87. Index. 91.

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(7) Chapter 0 Introduction 0.1. Historical background and motivation. Compared to other dimensions, the world of smooth 4-dimensional manifolds remains largely mysterious. Methods that offer many important answers in others dimensions do not have an adaptation applicable to dimension 4. An important breakthrough came towards the end of the last century with the introduction of the ideas of gauge theory from physics into mathematics. This breakthrough started with Donaldson’s theory [Don83], and continued with the inception of the Seiberg-Witten theory [Wit94]. The basic idea of the two theories is similar in principle. Both extract information about 4-manifolds by analysing the moduli space of solutions of certain differential equations on the manifold. The latter, however, has significant technical advantages over the former. As a result, the ensuing activity led to striking new insights in the world of smooth 4-dimensional manifolds ([Don96, §6], [Sco05, Ch. 10]). However, the Seiberg-Witten theory also has some limitations. One notable example is its inability to provide information about connected sums of 4-manifolds. A stable cohomotopy invariant was proposed and constructed in [BF04] as a new, non-mainstream way of describing the Seiberg-Witten invariant. Rather than directly analysing the moduli space of monopoles (i.e. solutions of the Seiberg-Witten equations), the central object of interest in [BF04] (as well as in its immediate successor [Bau04b]) is the so called monopole map. The idea of this new approach is to use the monopole map in a certain stable homotopy setup and construct a class associated to the underlying spinC 4-manifold. 1.

(8) Chapter 0. Introduction. 2. Since the Seiberg-Witten equations appear as the main ingredients of the monopole map, the resulting invariant is very closely related to the Seiberg-Witten invariant. It, however, yields some information on decomposable 4-manifolds undetected by the latter. The Seiberg-Witten theory is originally a theory for 4-dimensional manifolds. Its success in providing information about 4-manifolds motivated efforts to use it in the research of 3-dimensional manifolds. Adaptations to the 3-dimensional world were provided through several different approaches (notably [KM07, Nic03]). In this thesis, the possibility of applying the procedure from [BF04] to closed 3-dimensional manifolds is investigated. Although successful, direct application of the methods from [BF04] to the monopole map in three dimensions fails to yield interesting information about the underlying manifold. For this reason, instead of the monopole map, a family of a certain type of perturbations parametrised by the complex plane is analysed. The main focus of this thesis is the study of the limit behaviour of this family. In particular, a certain technical condition (the so called boundedness property) is proved. This condition is needed in order to be able to extend the new monopole maps to the 1-point compactification of a certain Hilbert space and is therefore vital for exploiting the stable homotopic apparatus used in [BF04].. 0.2. Thesis overview. Chapter 1 serves as a preparation for the main discussion in later chapters. Some general notational and computational conventions are covered in §1.1 and §1.2. Section 1.3 lists definitions of different scalar products and norms used in the concrete models for objects needed in the Seiberg-Witten theory which are presented in §1.4. These concrete models are defined using quaternions, which enable an elegant presentation of the objects in question and as a result simplify local calculations. In addition, some well-known related constructions are carried out explicitly in the given concrete setup. Sections 1.5 and 1.6 treat certain subspace of differential forms as a Clifford module and the corresponding Dirac operator defined on it is briefly discussed. The chapter’s raison d’ˆetre is Section 1.7, where a detailed analysis of the quadratic term is carried out. First, the quadratic term is described in terms of the quaternionic model from previous sections and a relation between some scalar products is established. Next, an expression for the derivative of the.

(9) 0.2. Thesis overview. 3. quadratic term is provided in §1.7.2. This expression plays an important role in Chapter 2 in the proof of the boundedness property. Moreover, it is used in the final part of the section to obtain an estimate of Sobolev norms of the quadratic term needed in the proof of the boundedness property in Chapter 3. In Chapter 2 the monopole map on a closed 3-dimensional manifold is defined along the lines of [BF04]. Its properties on a general closed 3-manifold are examined in detail, and its shortcomings are discussed. A modification of the monopole map is proposed in Chapter 3. Some interesting new terms are added and after suitable renormalisation, a family of monopole maps is obtained in §3.1. Section 3.2 deals with the boundedness property and is divided into three parts. An appropriate modification of the boundedness property for this family is introduced in §3.2.1. Subsection 3.2.2 shows how to obtain a priori estimates, and in §3.2.3 the estimates needed in the boundedness property are attained with the help of a modified version of the bootstrapping argument. The result, which is the main result of the thesis, is summarised in §3.3. After proving the modified boundedness property, a further renormalisation of the map is carried out in §3.4. Chapter 4 illustrates how the general discussion of the previous chapters applies to a concrete example: a 3-torus. The monopole map on a 3-torus is expliticly written down using concrete models and conventions from Chapter 1. This example was used to detect and understand subtle differences in the proofs of the boundedness property of different versions of the monopole map. Some general and well-known facts needed in the discussion of the monopole map are included in the Appendix. Section A.1 summarises selected important results from the theory of elliptic operators and Sobolev spaces. Due to a large amount of freedom at choosing the conventions and the inconsistency of the choice in the literature, several relatively basic and standard calculations were performed in the sections on Hodge theory (§A.2) and Clifford and exterior algebras (§A.3). Section A.4 recalls the definition of the connection Laplacian, and contains a basic calculation. Although their presence is not crucial, some auxiliary calculations are added with the intention of easing the readability. They appear in the form of endnotes in order to avoid lengthy digressions from the main text. Endnotes are indicated in the same way as footnotes, except the number is framed in order to distinguish the two. For example: an endnote 0 , a footnote0 . As a notational remark, throughout the thesis the letter Y is used to denote a closed 3-manifold, while M is reserved for a general closed n-manifold. Most of other notational conventions used in the text can be found in the list of notation..

(10) Chapter 0. Introduction. 4. The symbol J is used to indicate the end of remarks and conventions. Theorems, Propositions, Lemmas etc. are all numbered by the same counter to make them easier to find.. 0.3. A remark. Lastly, a remark regarding the proof of the boundedness property in Chapter 3. In the bootstrapping argument in §3.2.3, an estimate for the norm of the quadratic term and a similar estimate for Clifford product are used to carry the p argument through. However, the general fact that Sobolev Lk -completion is a Banach algebra for pk > n ([Pal68, Corollary 9.7]) is already enough to conclude the proof. At the time the proof of the boundedness property was being compiled, the above-mentioned fact about Sobolev spaces managed to escape my attention, and so a weaker version of Sobolev’s theorem (Theorem A.1.6) was used instead. Since using the weaker version only slightly prolongs the proof, the original version of the proof is left in the thesis. The shorter version would mean skipping §1.7.3 and the proof of (3.11), and using the above-mentioned fact to conclude the bootstrapping argument as explained in (appropriately located) Remark 3.2.4..

(11) Chapter 1 Preliminaries As mentioned in the introduction, this chapter will serve as a reminder of some results used later and also to fix notation and conventions used in the rest of the thesis. A significant part of the choice of conventions and models presented here is borrowed from [Bau12].. 1.1. Notation and remarks. The symbol . will denote inequalities up to a multiplication of the right-hand side by some positive constant. For example, if l − nq ≤ k − pn and l ≤ k holds, then instead of writing ∃ C = Cp,k,l,q > 0 s.t. k . k Lq ≤ C k . k Lp , l. k. we will write k . k Lq . k . k Lp . l. k. This notation will also be used to indicate that there is a bound on the set of objects (or rather on their norms). For example, in Chapters 2 and 3, an expression of the form ψ Lp . 1 will occur frequently. This means, that there exists k a fixed positive constant R 0 > 0 such that ψ Lp ≤ R 0 holds simultaneously for k all spinors ψ from some set we are interested in at the given moment. It is worth stressing at this point that the expression ψ Lp . 1 is not meant k p to indicate that ψ p < ∞ (i.e. that ψ is an L -spinor); rather, it means that its. p Lk -norm. Lk. k. is bounded by some fixed positive real number, as explained above. 5.

(12) Chapter 1. Preliminaries. 6. The following simple inequality will be repeatedly used in the text without mentioning it x + y p . kx k p + y p . (1.1) Here, k . k denotes an arbitrary norm (in the text usually one of the Sobolev norms, e.g. k . k Lp ) and p ≥ 0. This can easily be shown as follows       x + y p ≤ kx k + y p ≤ 2 max  kx k, y p ≤ 2p kx k p + y p .. 1.2. Some identifications and conventions. In this section we present a quaternionic model for spinors, Clifford bundle etc. The main motivation for presenting a concrete model for these objects will primarily be its use in the analysis of the quadratic term in §1.7. Also, it will be useful in describing relationships between scalar products of spinors as well as forms and endomorphisms (§1.3). R4 will be identified with the quaternions H in the obvious way R4 3 h = (x 0 , x 1 , x 2 , x 3 ) ≡ x 0 + ix 1 + jx 2 + kx 3 ∈ H.. (1.2). Specifically, R4 3 e 0 ≡ 1 ∈ H, R4 3 e 1 ≡ i ∈ H,. (1.3). R4 3 e 2 ≡ j ∈ H, R4 3 e 3 ≡ k ∈ H,. where e 0 , e 1 , e 2 , e 3 denotes the canonical basis of R4 . Generally, when using H as a model for the spinor bundle, we will denote it by ∆3 B H and call its elements Dirac spinors or simply spinors1 . The symbol ∆C3 will sometimes be used to stress the fact that we are interpreting ∆3 as a complex space (see (1.17)). Let e 1 , e 2 , e 3 ∈ R3 denote elements of the canonical ordered basis. R3 will be identified with the subspace of purely imaginary quaternions Im(H) via e 1 7→ i ∈ Im(H), 1. e 2 7→ j ∈ Im(H),. e 3 7→ k ∈ Im(H).. the notation and nomenclature are borrowed from [Fri97, p. 15]. (1.4).

(13) 1.3. Scalar products and norms. 7. The symbol Λ∗ (Rn ) will denote both the exterior algebra of Rn and the exterior algebra of its dual (Rn ) ∗ . Which interpretation will be used will depend on the context, and will generally not be mentioned explicitly. This imprecision is justified by the fact that Rn and (Rn ) ∗ are canonically isomorphic via the standard inner product. In situations where elements of Λ∗ (Rn ) are indexed, a lower index will suggest that the elements are interpreted as vectors (e.g. e 0 , . . . , en−1 ∈ Rn ), and an upper index will suggest that the elements are interpreted as covectors (e.g. e 0 , . . . , e n−1 ∈ (Rn ) ∗ ). The complexified exterior algebra of Rn and the complexified exterior algebra of its dual will be identified via the tensor product of the canonical isomorphism Rn  (Rn ) ∗ and the identity on C. This means, we will not identify the two complexified algebras via dualising with the help of the complexified inner product on them2 . In general, the Clifford algebra of Rn and the exterior algebra of Rn are canonically isomorphic (as vector spaces) via the assignment (cf. [LM89, p. 11]) Cn 3 eI 7−→ e I ∈ Λ∗ (Rn ),. (1.5). · ·. where I = (i 1 , . . . , ik ) is an ordered subset of (0, . . . , n − 1), eI B ei 1 . . . eik and e I B e i 1 ∧ . . . ∧ e ik . In accordance with the above-mentioned identification of the complexification of Λ∗ (Rn ) and its complexified dual, the complexified versions CnC = Cn ⊗R C and Λ∗C (Rn ) = Λ∗ (Rn ) ⊗R C will be identified via cl. CnC 3 eI ⊗ λ 7−→ e I ⊗ λ ∈ Λ∗C (Rn ),. cl. (1.6). i.e. via the vector space isomorphism (1.5) tensored with the identity on C. 1.2.1. Remark. The above isomorphisms will be understood without special mention when treating Clifford multiplication by a vector and its dual covector as the same operator. J. 1.3. Scalar products and norms. The scalar product. 0 h, h R B Re(h · h¯0 ), H. (1.7). as an example, we will have c (i ·S e 1 ) = c (i ·S e 1 ), and not c (i ·S e 1 ) = −c (i ·S e 1 ), where c denotes the Clifford multiplication with the corresponding element 2.

(14) Chapter 1. Preliminaries. 8. and the induced norm correspond under (1.2) to the standard Euclidean scalar product and the standard norm in R4 . The standard Hermitian product reads. 0 h, h C = Re(h · h¯0 ) − i Re(h · i · h¯0 ). H. H. H. (1.8). On the space EndH (∆3 ) = H of H-linear endomorphisms of ∆3 we define the following real inner product: ¯ ha, biEndH (∆3 ) B Re tr(ab ∗ ) = Re(ab).. (1.9). Complexification of (1.9) reads ¯ · λ¯ τ. ha ⊗ λ, b ⊗ τ i = Re(ab). (1.10). On the space EndC (∆3 ) of C-linear endomorphisms of ∆3 we define the following real inner product:   ha ⊗ λ, b ⊗ τ iEndC (H) B Re tr (a ⊗ λ) ◦ (b ⊗ τ ) ∗   = Re tr (a ⊗ λ) ◦ (b¯ ⊗ τ¯ )   ¯ · Re(λ¯ = Re tr (ab¯ ⊗ τ¯λ) = Re(ab) τ ). (1.11) When considering C-endomorphisms of ∆3 as complex matrices of order 2, we will be using hA, BiC(2) B. 1 Re tr(AB ∗ ), 2. (1.12). where A, B ∈ C(2), in line with the general definition hA, BiK(n) B. 1 Re tr(AB ∗ ), n. K ∈ {R, C, H}.. (1.13). If matrices A and B correspond3 to a ⊗ λ and b ⊗ τ respectively, the values of the scalar products coincide. In that way, the value of the scalar product is independent of the interpretation. The real scalar product (1.7) on H induces the standard Euclidean product on R3 through (1.4). This induces a scalar product on Λ∗ (R3 ) determined by the requirement  D I JE  1, I = J , e ,e B  (1.14)  0, I , J . ,  3. the correspondence will be specified in §1.4.1 (Conventions 1.4.6 and 1.4.10).

(15) 1.4. Clifford algebras and spin groups. 9. where I , J stand for strictly increasing ordered multi-indices I = (i 1 , . . . , ik ) and J = (j 1 , . . . , jl ), and e I stands short for e i 1 ∧ . . . ∧ e ik . In general, the standard inner product on Rn induces an inner product on Λ∗ (Rn ), and on Λ∗C (Rn ) = Λ∗ (Rn ) ⊗R C a Hermitian product by. α ⊗ λ, β ⊗ τ C B α, β · λ¯ τ, and an inner product by. τ ) = Re α ⊗ λ, β ⊗ τ C . α ⊗ λ, β ⊗ τ R B α, β · Re(λ¯ All these different inner and Hermitian products will be denoted by h. , .iR and h. , .iC respectively. Whenever possible, R will be omitted from h. , .iR to further simplify notation. Possible ambiguities will be left to the context to resolve. In places where the context is not clear enough, it will be explicitly mentioned which particular inner or Hermitian product is meant, or the appropriate suggestive notation will be used. Finally, note that for a 1-covector a ∈ Λ1 (R3 ) and a spinor h ∈ ∆3 we have D E D E ¯ = a, hih¯ End (∆ ) = a, hih¯ ∗ 3 . (1.15) hi · a h, hi∆3 = Re(ah¯ıh) Λ (R ) H 3 S. ·. cl. The last equality requires some explanation. Later in the chapter4 , we will specifiy a concrete isomorphism Λ0,1 (R3 )  EndC (∆3 ). With respect to the above C inner products, this isomorphism becomes an isometry (regardless of how one choses to write elements of the latter space). Also, it is worth pointing out that under the mentioned isometry, EndH (∆3 ) corresponds to the space Λ0,1 (R3 ).. 1.4. Clifford algebras and spin groups. b n2 c The general Dirac spinors5 ∆n = C2 for n = 3 yield ∆3 = C2 ≡ H. In particular, ∆3 = H is the unique irreducible real representation of Spin(3), and6 ∆3 ≡ C2 is the unique irreducible complex representation of Spin C (3) ([LM89, §I.5]). For this reason, ∆3 = H will serve as a local model for the spinor bundle, on which we will have a Clifford module structure, as well as the structure of a complex vector space. Considering the chosen quaternionic model, there are two main options for realisation of the two above-mentioned module structures: 4. see Convention 1.4.10 [Fri97, p. 15] 6 see (1.17) 5.

(16) Chapter 1. Preliminaries. 10. (i) Clifford product as the left-hand side H-multiplication, and scalar multiplication on the right-hand side; (ii) Clifford product as the right-hand side H-multiplication, and scalar multiplication on the left-hand side. Since both module structures naturally belong on the left-hand side, there is a certain amount of unnaturality in both cases. Thus, it is probably fair to say that it is a matter of taste which one of them one choses. In this thesis we will use the first option. 1.4.1. Convention. Left-hand side multiplication by a quaternion will represent Clifford multiplication, and right-hand side multiplication by a conjugate will represent scalar multiplication. J In order to avoid ambiguity with the Clifford product, a special symbol will sometimes be used to denote scalar multiplication. For a scalar λ (primarily from C), and a spinor h ∈ ∆3 = H we will write ¯ λ · h B h · λ, S. H. with · denoting quaternionic multiplication. As a consequence of Convention 1.4.1, we have H. 1.4.2. Convention. The space of quaternions, when considered as a 2-dimensional complex vector space, will be identified with C2 via x = x 0 + ix 1 + jx 2 + kx 3 ≡ (x 0 + ix 1 , x 2 − ix 3 ) = h 1 + jh 2 ≡ (h 1 , h 2 ).. (1.16). In other words, instead of the more natural (i.e. orientation preserving) identification H = C + Cj, we have H = C + j C ≡ C2 .. (1.17). This, furthermore, carries over to C(2) = EndC (C2 ) = R ⊕ su(2). Namely C(2) is now a complex vector space generated by the identity I and an antihermitian version {iσ3 , −iσ2 , −iσ1 } of trace-free Pauli matrices " # " # " # 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = . (1.18) 1 0 i 0 0 −1.

(17) 1.4. Clifford algebras and spin groups. 11. with the C-module structure given by f g λ · A ≡ A λ¯ ≡ x → 7 Ax · λ¯ .   As an example,f theg endomorphism A = 0i 0i ∈ C(2) acts on a vector H 3 v v 1 + jv 2 = v ≡ v12 ∈ C2 as follows: f v g f iv g Av = A v12 = iv12 ≡ v · i = −v · ı¯ = −i · v, S. H. S. H. hence A = −i · I .. J. S. 1.4.3. Remark. Following Convention 1.4.1, we have   C3C = C3 ⊗R C 3 a ⊗ u ≡ h 7→ a · h · u¯ . H. H. Every element of C3 can be written as a sum of elements of the form a ⊗u, with a ∈ C3 , u ∈ C. An element of SpinC (3) = Spin(3)×Z2 U (1) will be denoted by the suggestive symbol [a, u], with a ∈ Sp(1), u ∈ U (1) to stress that the pair (a, u) ∈ Sp(1) × U (1) is uniquely determined up to sign. J C. With the usual general definitions ([LM89, §I.4]), in dimension 3 we have the isomorphisms: C3 C3C Spin(3) Spin C (3).  H ⊕ H,  C(2) ⊕ C(2),  SU (2)  Sp(1),  U (2).. We will actually identify C3 with the subalgebra of diagonal matrices in H(2), and write C3 ⊆ H(2). Similarly, C3C will be identified with the subalgebra of C(4) consisting of block diagonal matrices, with blocks being elements of C(2). We proceed with making the above-mentioned identifications explicit. Convention (1.4) suggests the following assignment: " # " # 1 0 k 0 1 7→ ∈ H(2), e 1e 2 7→ ∈ H(2), 0 1 0 k " # " # i 0 −j 0 e 1 7→ ∈ H(2), e 1e 3 7→ ∈ H(2), 0 −i 0 −j " # " # (1.19) j 0 i 0 e 2 7→ ∈ H(2), e 2e 3 7→ ∈ H(2), 0 −j 0 i " # " # k 0 −1 0 e 3 7→ ∈ H(2), e 1e 2e 3 7→ ∈ H(2). 0 −k 0 1.

(18) Chapter 1. Preliminaries. 12. In short, we have: " # v 0 R ≡ im H 3 v 7→ ∈ H(2). 0 −v 3 (1.4). (1.20). 1.4.4. Remark. The minus sign in the lower right entry for e 1 , e 2 and e 3 serves to distinguish e 1 from e 2e 3 etc. J. Convention (1.17) implies the following correspondence i→ 7 iσ3 ∈ C(2), j→ 7 −iσ2 ∈ C(2), k→ 7 −iσ1 ∈ C(2),. (1.21). and hence we get the identifications for C3C ⊆ C(4): " # " # I 0 −iσ1 0 1 7→ ∈ C(4), e 1e 2 7→ ∈ C(4), 0 I 0 −iσ1 " # " # iσ3 0 iσ2 0 e 1 7→ ∈ C(4), e 1e 3 7→ ∈ C(4), 0 −iσ3 0 iσ2 " # " # −iσ2 0 iσ3 0 e 2 7→ ∈ C(4), e 2e 3 7→ ∈ C(4), 0 iσ2 0 iσ3 " # " # −iσ1 0 −I 0 e 3 7→ ∈ C(4), e 1e 2e 3 7→ ∈ C(4), 0 iσ1 0 I. (1.22). where, σ1 , σ2 , σ3 denote the Pauli matrices, as before (1.18). Using the universal property of Clifford algebras, it is easy to see that (1.20) H determines an isomorphism between abstractly defined C3 and ⊕ ⊆ H(2). H. C. The same applies to C3 and. C(2). ⊕ C(2). ⊆ C(4) via (1.22).. Given the above isomorphisms, we make the following (re)definitions: (" # ) H h 0 0 C3 B ⊕ = : h, h ∈ H ⊆ H(2), (1.23) 0 h0 H (" # ) C(2) h ⊗R u 0 C 0 0 ⊕ C3 B = : h, h ∈ H, u, u ∈ C ⊆ C(4). (1.24) 0 h 0 ⊗R u 0 C(2) Next we determine the subset of. H. which corresponds to the group Spin(3).. ⊕ H. General definition of the spin group reads Spin(n) = Pin(n) ∩ Cn0 , where Pin(n) ⊆ Cn×.

(19) 1.4. Clifford algebras and spin groups. 13. denotes the group generated by elements of the unit sphere S n−1 ⊆ Rn , and Cn0 denotes the +1-eigenspace of the automorphism of Cn induced by the assignment Rn 3 v 7→ −v ∈ Rn ([LM89, §I.2]). 1.4.5. Lemma. The group Pin(3) ∩ C30 corresponds to the group (" # ) a 0 G= : a ∈ Sp(1) . 0 a under the identification (1.19). Proof. First note that ( ) Pin(3) ∩ C30 = ±1, q 1q 2 , : q 1 , q 2 ∈ S 2 ⊆ R3 , ( ) = q 1q 2 , : q 1 , q 2 ∈ S 2 ⊆ R3 . Let G 0 ⊆ H(2) denote the image of Pin(3) ∩ C30 under the isomorphism (1.19) P3 P and chose arbitrary q 1 = i=1 λi ei ∈ S 2 and q 2 = 3j=1 µ j e j ∈ S 2 . The product q 1q 2 takes the form q 1q 2 = −. 3 X i=1. λi µ i +. X. (λi µ j − λ j µi )ei e j .. i<j. The quaternion that emerges after applying (1.19) has the norm equal 1 to 1. Hence, it is clear that G 0 is a subgroup of G. The adjoint representation of the spin group7 determines a double covering Spin(3) → SO (3). Thus dim G 0 = dim Spin(3) = dim SO (3) = 3 = dim Sp(1) = dim G. Since G is connected, there follows G 0 = G.  Therefore, as previously with the Clifford algebras, we set (" # ) a 0 Spin(3) B ∈ H(2) : a ∈ Sp(1) , 0 a (" # ) [a, u] 0 C Spin (3) B ∈ C(4) : a ∈ Sp(1), u ∈ U (1) . 0 [a, u] 7. see (1.30).

(20) Chapter 1. Preliminaries. 14. Obviously, C3 ⊆ C3C and also Spin(3) ⊆ Spin C (3). One element of the real Clifford algebra Cn of Rn will play an important role later. It is the so called volume element ω B e 0 · · · en−1 ∈ Cn ,. (1.25). where e 0 , . . . , en−1 denote the vectors of the canonical basis in Rn . The complex analogue of the real volume element (1.25) is defined as ωC B i b. n+1 2. − b n+1 c ·e · · ·e 2 c ∈ C C, 0 n−1 = e 0 · · · en−1 ⊗ i n S. (1.26). where e 0 , . . . , en−1 denote the vectors of the canonical basis in Cn . It will be called the complex volume element 8 . Definitions (1.25) and (1.26) do not depend on the choice of the oriented orthonormal basis of Rn . Note that, under (1.22), the complex volume element in C3C takes the form ωC = i b. 1.4.1. n+1 2. " c · e e e = −e e e = 1 2 3 1 2 3. (1.22). S. # I 0 ∈ C(4). 0 −I. (1.27). Spin representation. 1.4.6. Convention. In (1.19) and (1.22), only the upper left entry acts on ∆3 , on which the spinor bundle S is modelled. This yields a map C3C → EndC (∆3 ) = C(2). For example, this in particular means that e 1 ∈ C3C and e 2e 3 ∈ C3C represent the same endomorphism. The same goes for the corresponding covectors. For example, e 1 and e 2 ∧ e 3 = ∗e 1 represent the same endomorphism on ∆3 . More concisely, as suggested by (1.27), an element e ∈ C3C and its ”dual” ωC e determine the same endomorphism of ∆3 . Using the relation (A.12) between ωC and the Hodge star operator, the same holds for covectors after applying the canonical isomorphism C3C  Λ∗C (R3 ) (1.5). J. ·. cl. The spin representation is defined by ρ : Spin(3) → EndH (∆3 ) = H, " #   a 0 7→ h → 7 a ·h . 0 a H. 8. see [LM89, p. 34]. (1.28).

(21) 1.4. Clifford algebras and spin groups. 15. and analogously, the spinC representation by ρ C : SpinC (3) → EndC (∆3 ) = C(2), " # [a, u] 0 ¯ 7→ (h 7→ a · h · u). 0 [a, u] H. (1.29). H. 1.4.7. Remark. Up to isomorphism, there are two different irreducible real representations of C3 ([LM89, §I.5]). The choice in Convention 1.4.6 of the entry which actually acts on spinors corresponds to the choice of irreducible representation. J. The adjoint representation of Spin(3) is given by Ad : Spin(3) → EndR (im H) ≡ EndR (R3 ) = R(3), " # a 0 ¯ 7→ (h 7→ a · h · a). 0 a H. (1.30). H. This definition is motivated by the fact that the assignment in (1.30) determines a double covering 2 Sp(1) → SO (3). The adjoint representation of SpinC (3) reads:. ". Ad : Spin C (3) → EndC (im H ⊗R C) ≡ C(3), #   (1.31) [a, u] 0 ¯ ⊗ (u · λ · u) ¯ = (a · h · a) ¯ ⊗λ . 7→ (h ⊗ λ) 7→ (a · h · a) 0 [a, u] H. H. H. H. H. H. Lastly, the Clifford multiplication on ∆3 is given by   cl : im HAd ⊗R C ⊗R (∆3 )ρ C → (∆3 )ρ C ¯ (v ⊗ λ) ⊗ h → −v¯ · h · λ¯ = v · h · λ, H. H. H. (1.32). H. or in terms of matrices " # v ⊗λ 0 im H ⊗ C 3 v ⊗ λ 7→ ∈ C(4), 0 −v ⊗ λ. (1.33). with Convention 1.4.6 in mind. Obviously, (1.32) is a homomorphism of representations, i.e.      C ¯ · (a · h · u) ¯ · λ¯ ρ ([a, u]) cl (v ⊗ λ) ⊗ h = a · v · h · λ¯ · u¯ = (av a)    C = cl Ad(a)(v) ⊗ λ ⊗ ρ ([a, u])(h) . (1.34) H. H. H. H. H. H. H. H.

(22) Chapter 1. Preliminaries. 16. 1.4.8. Remark. Note the compatibility of (1.20) and (1.33).. J. 1.4.9. Remark. The above definition of the Clifford multiplication (1.32), (1.33) (using left-hand side quaternion multiplication) was the main motivation behind identifications (1.16). J. ·. When needed, we will use the symbol to denote the above defined Clifford action on ∆3 . The same symbol will also be used for the algebra operation in CnC . cl. 1.4.10. Convention. Convention 1.4.6 yields a map C3C → EndC (∆3 ) = C(2). However, it will be important to be able go in the other direction too. I.e. it will be important to interpret endomorphisms of ∆3 as covectors or elements of C3C .. For that, we identify EndC (∆3 ) = C(2) with the subspace 1, e 1 , e 2 , e 3 C ⊆ C3C via combination of assignments (1.4) and (1.21), i.e. 1 7→ I ∈ C(2), e 1 7→ iσ3 ∈ C(2), e 2 7→ −iσ2 ∈ C(2), e 3 7→ −iσ1 ∈ C(2).. (1.35). This induces a vector space isomorphism between the above-mentioned spaces, and its inverse will serve as a translation of endomorphisms into covectors9. using the fact that 1, e 1 , e 2 , e 3 C ⊆ C3C and Λ0,1 (R3 ) are isomorphic via the C (1.5). canonical isomorphism C3C  Λ∗C (R3 ).. J. Before moving on, we mention one more lemma10 : 1.4.11. Lemma. The Pin(2) group corresponds to the normaliser of U (1) in Sp(1) and equals Pin(2) = U (1) t j U (1). Sketch of proof. In general, we have C2  H and C2C  C(2) ([LM89, §I.4]). So, C2 and C2C can be seen as a half of C3 and C3C respectively. Similarly as in the proof of Lemma 1.4.5 we have ( ) Pin(2) = ±1, q 1 , q 1q 2 : q 1 , q 2 ∈ R2 , q 1 = q 2 = 1 9. in particular, this will be applied in the second Seiberg-Witten equation, to the quadratic. term 10. we will use this lemma in Proposition 2.3.13.

(23) 1.4. Clifford algebras and spin groups. 17. ) ) ( ( = q 1 : q 1 ∈ R2 , q 1 = 1 t q 1q 2 : q 1 , q 2 ∈ R2 , q 1 = q 2 = 1 . Assign (1, e 1e 2 , e 1 , e 2 ) 7→ (1, i, j, k ) ∈ H4 . The second part is now obviously isomorphic to U (1). An element q 1 from the first set is clearly of the form q 1 = e 1q01 , with q01 being an element of the second set. The above group is clearly contained in the normaliser N of U (1) in Sp(1). Conversely, let q = a + jb ∈ N , with a, b ∈ C. Since, ¯ − ba¯u) ¯ ∈ U (1), quq¯ = |a| 2u + |b| 2u¯ + j (bau. ∀u ∈ U (1),. there follows ba¯ = 0, i.e. either q ∈ U (1) or q ∈ jU (1).. 1.4.2. . Other representations of the spin group. For Spin C (3) there are following short exact sequences of groups11 : ι. κ. 1 −→ U (1) −→ SpinC (3) −→ SO (3) −→ 1, ι. (1.36). l. 1 −→ Spin(3) −→ SpinC (3) −→ U (1) −→ 1.. (1.37). Here, l : Spin C (3) → U (1) is given by l : [a, u] 7→ u 2 and κ is the composition of the double covering map Spin C (3) → SO (3) × U (1) determined by the double covering12 Spin(3) → SO (3) and the map l and the projection onto the first factor.. 1.4.3. The associated bundles. For a closed oriented smooth 3-dimensional Riemannian manifold Y equipped C with a spinC structure s (i.e. a certain principal SpinC (3)-bundle P Spin lifting the frame bundle P SO ), the spinor bundle is defined as the bundle S associated C to P Spin via the spin representation (1.29): S B P Spin ×ρ C ∆3 . C. (1.38). Its sections will be referred to as spinors and typically denoted by ψ or ϕ. Note that spin representation (1.28) is a special orthogonal representation on ∆3 with 11 12. [Fri97, p. 28] discussed on page 75.

(24) Chapter 1. Preliminaries. 18. respect to scalar product (1.7), i.e. ρ : Spin(3) → SO (3). Similarly, spinC representation (1.29) is a unitary representation on ∆3 with respect to Hermitian product (1.8), i.e. ρ C : Spin C (3) → U (2). Hence, spinor bundle (1.38) comes with a natural Hermitian product. 1.4.12. Remark. In the case of spin structure, the spinor bundle is quaternionic line bundle and the Dirac operator preserves the quaternionic structure . J. The adjoint representation (1.31) can be used to define the tangent and cotangent bundles of Y : TY B P Spin ×Ad Im H, C. T ∗Y B P Spin ×Ad Im H, C. and their complexifications TCY B P Spin ×Ad (Im H ⊗R C), C. TC∗Y B P Spin ×Ad (Im H ⊗R C). C. The (local) identification convention from §1.2 carries over to the present context of complexified tangent and cotangent bundles. The representation l : Spin C (3) → U (1) from (1.37) gives rise to the determinant bundle of s: C L = det s = P Spin ×l C. To the frame bundle P SO we can associate the so called Clifford bundle C3 (Y ) B P SO ×cl(ι) C3 ,. (1.39). via the Clifford representation13 cl(ι) : SO (3) → Gl (C3 ), f g cl(ι)(A) (v 1 · . . . · vk ) = ι(A)v 1 · . . . · ι(A)vk = Av 1 · . . . · Avk .. (1.40). Here, ι : SO (3) → Gl (3, R) denotes the inclusion. The representation Ad : Spin(3) → Gl (C3 ), ¯ Ad(a)(v 1 · . . . · vk ) = a · v 1 · . . . · vk · a¯ = av 1a¯ · . . . · avk a, clearly descends (using the double covering Spin(3) → SO (3)) to a representation of SO (3), which coincides14 with cl(ι) in (1.40). Therefore, C3 (Y ) can also 13 14. cf. [LM89, p. 95] cf. [LM89, p. 96].

(25) 1.5. A different Clifford module structure on forms. 19. be defined by C3 (Y ) = P Spin ×Ad C3 .. (1.41). The complex Clifford bundle is defined analogously as in (1.41) C3C (Y ) = P Spin ×Ad C3C . C. (1.42). Due to (1.34), Clifford multiplication (1.32) induces well-defined bundle maps cl : TCY ⊗ S → S, cl : TC∗Y ⊗ S → S,. (1.43) (1.44). cl : C3C (Y ) ⊗ S → S.. (1.45). and from there also the map. It is easy to show that all the other local discussions from this chapter carry over to the corresponding bundles by a similar argument. Note that with the identifications from §1.2 in mind, the maps (1.43) and (1.44) become equal.. 1.5. A different Clifford module structure on forms. In the case n = 3, the subspace Ω1,0 (Y ) B Ω 1 (Y ) ⊕ Ω0 (Y ) ⊆ Ω∗ (Y ) of forms of degrees 1 and 0 will be of particular interest in the discussion of the monopole map. On it we can define a slightly different Clifford module structure than usual (cf. §A.3):   v ∗ · α,  c (v) = v α B  ∗  ∗(v ∧ α ) − ι(v)α,. ·. cl. α ∈ Λ0 (R3 ), α ∈ Λ1 (R3 ).. (1.46). Clearly, this defines a Clifford module structure on Λ1,0 (R3 ) B Λ1 (R3 ) ⊕Λ0 (R3 ), which carries over to Ω 1,0 (Y ). This modification is motivated by the fact that c (α ) = c (∗α ) for all α ∈ 1 Ω (Y ). Namely, an important consequence of (1.27), (A.12) and Convention 1.4.6 is c (α ) = c (∗α ) ∈ EndC (∆3 ), ∀α ∈ Λ1 (R3 ). (1.47).

(26) Chapter 1. Preliminaries. 20 On the level of spinors and 1-forms on Y , this means c (α ) = c (∗α ) ∈ EndC (S ),. ∀α ∈ Ω1 (Y ).. (1.48). In other words, Clifford multiplication by 2-forms does not introduce new endomorphisms of S, so by staying in the space Ω 1,0 (Y ) nothing is lost. Therefore, defining a Clifford module structure on the space Ω1,0 (Y ) makes sense.. ·. 1.5.1. Remark. Note that (1.46) implies e 1 e 2 = e 3 etc. which via15 (1.4) translates into standard relations between i, j, k ∈ H. J cl. 1.6. Dirac operator on forms. In general, the usual Clifford module structure16 defines together with the extension of the Levi-Civita connection to Ω ∗ (M ) a Dirac operator which equals17 the Hodge-de Rham operator D HdR = d + d ∗ : Ω∗ (M ) → Ω∗ (M ). In the case of a 3-manifold Y , the following slight modification the Hodgede Rham operator on Ω1,0 (Y ) = Ω 1 (Y ) ⊕ Ω0 (Y ) appears in the later discussion of the monopole map: # ∗d d DΩ B ∗ : Ω1,0 (Y ) → Ω1,0 (Y ). d 0 ". (1.49). The Levi-Civita connection on Ω∗ (Y ), together with the Clifford module structure (1.46) determines a Dirac operator on Ω1,0 (Y ) which actually equals D Ω . Thus, D Ω is an elliptic operator. The fact that D Ω is elliptic also follows directly from ". ∗d d d∗ 0. #2. # " ∗ # d d + dd ∗ 0 ∗d ∗d + dd ∗ ∗d 2 = = = (d + d ∗ ) 2

(27)

(28)

(29) Ω1,0 (Y ) . 0 d ∗d d ∗ ∗d d ∗d ". Here, keep in mind that18 ∗d ∗

(30)

(31)

(32) Ω2 (Y ) = d ∗

(33)

(34)

(35) Ω2 (Y ) . and with the use of the canonical isomorphism R3  (R3 ) ∗ see §A.3 17 [LM89, §II.6] 18 see (A.8) on p. 72 15. 16.

(36) 1.7. The quadratic term. 1.7. 21. The quadratic term. The quadratic term is most commonly defined as an endomorphism of S and then interpreted as a differential form using the inverse of (1.35) from Convention 1.4.10. For ψ = ψ 0 + jψ 1 ∈ ∆3 we define σ (ψ ) ∈ EndC (∆3 ) by 1 σ (ψ ) B ψ ⊗ ψ ∗ − |ψ | 2I 2 . 2. (1.50). Writen as a matrix, the above endomorphism takes the following form " # 1 |ψ 0 | 2 − |ψ 1 | 2 2ψ 0ψ¯1 σ (ψ ) = · 2ψ¯0ψ 1 |ψ 1 | 2 − |ψ 0 | 2 2 1 (1.18) = Re(ψ¯0ψ 1 )σ1 + Im(ψ¯0ψ 1 )σ2 + (|ψ 0 | 2 − |ψ 1 | 2 )σ3 ∈ C(2). 2 Using (1.22) together with Convention 1.4.6 that ei acts on ψ with the upper left 2 × 2 complex matrix we get: 1 σ (ψ ) = ψ ⊗ ψ ∗ − |ψ | 2I 2 = 2 1 = Re(ψ¯0ψ 1 )σ1 + Im(ψ¯0ψ 1 )σ2 + (|ψ 0 | 2 − |ψ 1 | 2 )σ3 2 1 = Re(ψ¯0ψ 1 )(−iσ1 ) · i + Im(ψ¯0ψ 1 )(−iσ2 ) · i + (|ψ 0 | 2 − |ψ 1 | 2 )iσ3 · (−i) 2 1 = −i · Re(ψ¯0ψ 1 )e 3 + −i · Im(ψ¯0ψ 1 )e 2 + i · (|ψ 0 | 2 − |ψ 1 | 2 )e 1 ∈ EndC (∆3 ), 2 S. S. S. which under (1.6) corresponds to the 1-covector 1 σ (ψ ) = −i · Re(ψ¯0ψ 1 )e 3 + −i · Im(ψ¯0ψ 1 )e 2 + i · (|ψ 0 | 2 − |ψ 1 | 2 )e 1 . 2 S. S. S. (1.51). The analogous claim holds for a spinor ψ ∈ Γ(S ). I.e. for a spinor ψ ∈ Γ(S ), its square 1 σ (ψ ) B ψ ⊗ ψ ∗ − |ψ | 2I , (1.52) 2 is a well-defined section of EndC (S ), with I denoting the identity map on S. This section can be interpreted as is an imaginary-valued 1-form on Y in the same way as above. The local expression of this form is analogous to (1.51): 1 σ (ψ ) = −i · Re(ψ¯0ψ 1 )e 3 + −i · Im(ψ¯0ψ 1 )e 2 + i · (|ψ 0 | 2 − |ψ 1 | 2 )e 1 . 2 S. S. S. (1.53).

(37) Chapter 1. Preliminaries. 22. 1.7.1. Two ways of writing the quadratic term. There are two common ways of writing the quadratic term in the literature on Seiberg-Witten theory. Definition (1.52) is the most common, but we will find it more convenient to use a slightly different definition. The quadratic term σ (ψ ) is related to the following quadratic map19 q(ψ ) = ψ i ψ¯ ∈ EndC (S ),. (1.54). which is acting on spinors by quaternionic multiplication on the left-hand side. To see this, write locally ψ = ψ 0 + jψ 1 and q(ψ ) = ψiψ¯ = (ψ 0 + jψ 1 )i (ψ¯0 − jψ 1 ) = i (

(38)

(39) ψ 0

(40)

(41) 2 −

(42)

(43) ψ 1

(44)

(45) 2 ) + 2jiψ¯0ψ 1 = i (

(46)

(47) ψ 0

(48)

(49) 2 −

(50)

(51) ψ 1

(52)

(53) 2 ) + 2jiRe(ψ¯0ψ 1 ) − 2jIm(ψ¯0ψ 1 ). Together with (1.4) we locally have q(ψ ) = (

(54)

(55) ψ 0

(56)

(57) 2 −

(58)

(59) ψ 1

(60)

(61) 2 )e 1 − 2Re(ψ¯0ψ 1 )e 3 − 2Im(ψ¯0ψ 1 )e 2 ∈ EndC (S ), and now it follows from (1.53) that i · q(ψ ), 2 The following calculation confirms the above identity: σ (ψ ) =. S. (1.55).  1 1 1 1 i · (q(ψ ) ψ ) = q(ψ ) ψ ı¯ = ψ · i · ψ¯ · ψ · ı¯ = |ψ | 2ψ = σ (ψ )ψ = σ (ψ ) ψ . 2 2 2 2 S. ·. ·. cl. cl. H. H. H. H. ·. cl. In analogy to (1.15) we have for sections of the corresponding bundles the simple but important identity. ·. ψ , iα ψ S = α, q(ψ ) EndC (S) = α, q(ψ ) Ω = α, −2iσ (ψ ) Ω . cl. (1.56). The symbol h . , . iΩ denotes the scalar product on forms, and h . , . iS denotes the real part of the Hermitian product on S. 19. this quadratic map is well-defined because the local definition carries over to bundles, due to commutativity of the appropriate representations (§1.4.1): q(ρ C ([a, u])(h)) = ¯ H· a¯ = Ad(a)(q(h)) ¯ H· i H· (a H· h H· u) ¯ = a H· (hih) (a H· h H· u).

(62) 1.7. The quadratic term. 1.7.2. 23. Derivation of the quadratic term. Later in the estimates, a close-up analysis of the derivation of the quadratic term will be needed. We include it at this point in the form of the present subsection in order not to disturb the flow later. Let (e 1 , e 2 , e 3 ) be a local orthonormal frame on Y centred at some arbitrary fixed point y0 ∈ Y (i.e. all Christoffel symbols vanish at y0 ). Unless specified otherwise, all calculations in this section will be done locally, at point y0 . P3 es (∇A )es ψ , and at y0 the following Locally, DA is of the form DAψ = s=1 20 holds :. ·. cl. 3. ·. E. XD DAψ , iψ = es (∇A )es ψ , iψ cl. s=1. =− =− =− =− =−. 3 D X s=1 3 X. (∇A )es ψ , iesψ. E 3. ·. E. XD ∂s ψ , iesψ + ψ , i (∇A )es (es ψ ) cl. s=1 3 X s=1 3 X s=1 3 X. ∂s ψ , iesψ +. s=1 3 D X. 3 D X E ψ , ies (∇A )es ψ ψ , i (∇es es ) ψ ) +. ·. E. cl. s=1. ·. cl. s=1. ∂s ψ , iesψ + 0 + ψ , iDAψ. . ∂s ψ , iesψ − DAψ , iψ .. s=1. P3 ∂s ψ , iesψ . The latter sum, however, equals 12 d ∗q(ψ ), I.e. DAψ , iψ = − 12 s=1 due to 3 3 X. (1.56) X. ∂s es , q(ψ ) Ω = ∗d ∗q(ψ ) ∂s ψ , iesψ = s=1. s=1. and the fact that d ∗ : Ω1 (Y ) → Ω 0 (Y ) equals21 − ∗d ∗. In other words. 1 (1.55) DAψ , iψ = d ∗q(ψ ) = −id ∗σ (ψ ). 2 20 21. cf. [LM89, p. 115] see (A.8). (1.57).

(63) Chapter 1. Preliminaries. 24. On the other hand, for an arbitrary 1-form a and its local expression a = r 22 r =1 ar e we have at y 0. P3. 3 E. XD DAψ , ia ψ = es (∇A )es ψ , iar er ψ. ·. cl. ·. ·. cl. =. r,s=1 3 D X. cl. 3 D E X E es (∇A )es ψ , iar er ψ + es (∇A )es ψ , iar er ψ. ·. ·. cl. r,s=1 r =s. ·. cl. ·. cl. r ,s=1 r ,s. cl. 3 E. XD = (∇A )aψ , iψ + es (∇A )es ψ , iar er ψ .. ·. ·. cl. r ,s=1 r ,s. cl. Simple calcuations 3 D 3 D X X E E es (∇A )es ψ , iar er ψ = − (∇A )es ψ , iar es er ψ. r,s=1 r ,s. =−. =−. =−. =−. 3 X r ,s=1 r ,s 3 X r ,s=1 r ,s 3 X r ,s=1 r ,s 3 X. ∂s ψ , iar es er ψ +. ∂s ψ , iar es er ψ +. r ,s=1 r ,s 3 D X. r,s=1 r ,s 3 X.  E ψ , i (∇A )es ar es er ψ. 3 E. XD ψ , i (∂s ar )es er ψ + ψ , iar es er (∇A )es ψ. r,s=1 r ,s 3 X. f. g ∂s ar ψ , ies er ψ +. ar · ∂s ψ , ies er ψ −. r ,s=1 r ,s. r ,s=1 r ,s 3 D X. ∂s ar ψ , ies er ψ −. r,s=1 r ,s 3 D X r,s=1 r ,s. r,s=1 r ,s. ψ , iar er es (∇A )es ψ. E. E iar er ψ , es (∇A )es ψ ,. imply 3 3 D X E. 1X ar · ∂s ψ , ies er ψ , es (∇A )es ψ , iar er ψ = − 2 r ,s=1 r,s=1 r ,s. 22. r ,s. Clifford multiplication by a 1-form is, of course, to be understood as the Clifford multiplication by the corresponding dual vector field, with the appropriate convention from §1.2 in mind.

(64) 1.7. The quadratic term. 25. and thus 3. . 1X ar · ∂s ψ , ies er ψ . DAψ , iaψ − (∇A )aψ , iψ = − 2 r ,s=1 r ,s. For the last sum, note that. ·. (1.56). ψ , ies er ψ = es er , q(ψ ) EndC (S ) . cl. According to Convention23 1.4.6, the elements es er ∈ C3C and their ”Hodge duals”24 ω Ces er ∈ C3C determine the same endomorphisms of ∆3 . Using the map EndC (∆3 ) → Λ0,1 (R3 ) from Convention 1.4.10, we conclude that the endomorC phism es er corresponds to the 1-covector ∗(e s ∧ e r ). Consequently,. ∂s ψ , ies er ψ = ∂s ∗(e s ∧ e r ), q(ψ ) .. ·. cl. Inserting this into the above sum leads to 3 X. 3 X. ar · ∂s ψ , ies er ψ = ar · ∂s ∗(e s ∧ e r ), q(ψ ). ·. cl. r ,s=1 r ,s. =. r,s=1 r ,s 3 X. ar ·. r =1. =. 3 X. ∂s ∗(e s ∧ e r ), q(ψ ) s=1 r ,s. .  a 1 · − ∂2 [q(ψ )]3 + ∂3 [q(ψ )]2   + a 2 · + ∂1 [q(ψ )]3 − ∂3 [q(ψ )]1   + a 3 · − ∂1 [q(ψ )]2 + ∂2 [q(ψ )]1 ,. with the last sum equalling 3 a, − ∗dq(ψ ) . Symbols [q(ψ )]l denote local component functions of the 1-form q(ψ ). In short, we get. . 1. DAψ , ia ψ − (∇A )aψ , iψ = a, ∗dq(ψ ) . 2. ·. cl. 23 24. see also (1.27) see (A.12) for the reason behind this name.

(65) Chapter 1. Preliminaries. 26. Finally, summarising the above calculations brings (1.49). D Ωq(ψ ) = (∗d + d ∗ )q(ψ ) 3 X. =2 DAψ , ies ψ e s − 2 ∇Aψ , iψ + 2 DAψ , iψ .. ·. cl. s=1. (1.58). Note that the first two summands in the last expression are 1-forms, and the third one is a function. Also, if a + f ∈ Ω1,0 (Y ), the above formula implies 3. X. D Ωq(ψ ), a + f = 2 DAψ , i (a + f )ψ − 2 (∇A )a∗ψ , iψ .. (1.59). s=1. 1.7.3. Norm of the quadratic term. From (1.53) follows the pointwise equality 1 |σ (ψ )| 2 = Re(ψ¯0ψ 1 ) 2 + Im(ψ¯0ψ 1 ) 2 + (|ψ 1 | 2 − |ψ 0 | 2 ) 2 4 1 1 1 2 2 2 2 = |ψ¯0ψ 1 | + (|ψ 1 | − |ψ 0 | ) = (|ψ 1 | 2 + |ψ 0 | 2 ) 2 = |ψ | 4 , 4 4 4. (1.60). and so iσ (ψ ) 2 = L. Z. ! |σ (ψ )| dvol 2. Y. 1 2. =. Z Y. 1 4 |ψ | dvol 4. !. 1 2. =. 1 2 ψ 4. 2 L. For the derivative of q(ψ ), the equation (1.58) implies

(66)

(67) D Ωq(ψ )

(68)

(69) ≤ 2

(70)

(71) DAψ

(72)

(73)

(74)

(75) ψ

(76)

(77) + 2

(78)

(79) ∇Aψ

(80)

(81)

(82)

(83) ψ

(84)

(85) + 2

(86)

(87) DAψ

(88)

(89)

(90)

(91) ψ

(92)

(93) ,. (1.61). and thus D Ωq(ψ ) 2 ≤ 2 DAψ 2 ψ 0 + 2 ∇Aψ 2 ψ 0 + 2 DAψ 2 ψ 0 L L C L C L C . ψ L21 ψ C 0 . The identity (1.58) further helps in estimating higher derivations Dmq(ψ ) of q(ψ ), for m ≥ 1. Namely, (1.58) indicates that higher derivatives of the quadratic term involve taking repeated exterior derivatives and coderivatives . of DAψ , ies ψ , ∇Aψ , iψ and DAψ , iψ . Using the fact that ∇A is a metric. ·. cl.

(94) 1.7. The quadratic term. 27. connection, we conclude that the component functions of Dmq(ψ ) take one of the following forms   (∇A )er . . . (∇A )er DAψ , ies (∇A )er . . . (∇A )er ψ , s+1 1 s  t −1 or (∇A )er . . . (∇A )er ψ , ies (∇A )er . . . (∇A )er ψ , 1 s s+1 t   or (∇A )er . . . (∇A )er DAψ , i (∇A )er . . . (∇A )er ψ ,. ·. cl. ·. cl. s. 1. s+1. t. where s + t = m and r j ∈ {1, 2, 3}. This leads to a pointwise inequality similar to (1.61) X X

(95)

(96) Dmq(ψ )

(97)

(98) .

(99)

(100) ∇s DAψ

(101)

(102)

(103)

(104) ∇t−1ψ

(105)

(106) +

(107)

(108) ∇s ψ

(109)

(110)

(111)

(112) ∇t ψ

(113)

(114) . (1.62) A A

(115) A

(116)

(117) A

(118) s,t ≥0 s+t=m. s,t ≥0 s+t=m. Integration gives Dmq(ψ ) 2 . Ω L. X s,t >0 s+t=m. ∇s DAψ 2 ∇t−1ψ 2 + A L A. L. ψ 0 + + ∇m−1 A D Aψ L2 C . ∇m A ψ L2 ψ C 0 +. X s,t >0 s+t=m. L. ∇s ψ 2 ∇t ψ 2 A L A L. s,t >0 s+t=m ∇mψ 2 ψ 0 A L C. ∇s ψ 2 ∇t ψ 2 , A L A L. and from that it directly follows for m ≥ 1 Dmq(ψ ) 2 . ψ 2 ψ Ω. X. Lm. C0. + ψ L2 2 .. (1.63). m. For m = 0 we simply have q(ψ ) L2 . ψ L2 ψ C 0 . Thus m X D j q(ψ ) q(ψ ) 2 . Lm Ω L2 j=0. . ψ L2 ψ C 0 +. m X j=1. ψ 2 ψ 0 + ψ 2 2 Lj C Lj. . ψ Lm2 ψ C 0 + ψ L2 2 . m In short q(ψ ) 2 . ψ 2 ψ 0 + ψ 2 2 , Lm Lm C Lm. m ≥ 0.. Obviously, the same inequality holds for σ (ψ ) due to (1.55).. (1.64).

(119)

(120) Chapter 2 The monopole map on 3-manifolds 2.1. Assumptions and general context. Unless stated otherwise, Y will denote a closed 3-manifold, and the following will be assumed: • Y is oriented and equipped with a Riemannian metric. • Y is equipped with a spinC structure s. • On the determinant bundle L B det s of the chosen spinC structure s, a unitary connection A is fixed such that [FA ] = −2πi c 1 (s). This gives a 1-1 correspondence Conn(L)  iΩ1 (Y )  Ω1 (Y ). If the spinC structure given above is a spin structure, we take A to be the trivial connection (as the natural choice). • A point y0 ∈ Y will be fixed.. 2.2. Seiberg-Witten equations on 3-manifolds. After having fixed a spinC structure on Y , all the definitions and constructions needed for writing down the Seiberg-Witten equations on closed 4-dimensional manifolds can be carried out in the 3-dimensional case. Thus we are able to write down the Seiberg-Witten equations for (Y , s): DA+ia (ψ ) = 0, ∗FA+ia − σ (ψ ) = 0. 29. (2.1a) (2.1b).

(121) Chapter 2. The monopole map on 3-manifolds. 30. The analysis of these equations proceeds in several different directions (e.g. [KM07, Chapter 4], [Nic03]). Here, they will be used to define the monopole map for 3-manifolds.. 2.3. Monopole map for closed 3-manifolds. The procedure presented here follows the construction from [BF04].. 2.3.1. Definition. In [Fri97, p. 189] the following theorem is stated: 2.3.1. Theorem. (Weyl’s Theorem) Let P be some principal U (1)-bundle over a 2 compact n-manifold M with the first Chern class c 1 (P ) ∈ HdR (M; R), and set ( ) 2 F (P ) B ω ∈ Ω (Y ) : dω = 0, [ω] = c 1 (P ) . Then the quotient Ψ : Conn(P )/G(P ) → F (P ) 1 FA is surjective, with each fibre of the map taking a connection A on P to − 2πi diffeomorphic to the Picard manifold Pic(M ) = H 1 (M; R)/H 1 (M; Z) of M.. Consider the fibre corresponding to the chosen connection A. That is to 1 say, we consider the preimage of − 2πi FA under the map Conn(P ) → F (P ), 0 1 0 A 7→ − 2πi FA , which actually equals the space A + i ker d of all connections on L having the same curvature as A. Division by G yields the Picard torus, as the above theorem states. In order to have a free action on A + i ker  d, we will restrict our attention to the action of G0 B ker evy0 : G → U (1) , where evy0 denotes the evaluation map at the chosen point y0 . Set   A˜ (Y ) B (A + i ker d ) × Γ(S ) ⊕ Ω 1 (Y ) ⊕ Ω0 (Y ) ,   ˜ ) B (A + i ker d ) × Γ(S ) ⊕ Ω 1 (Y ) ⊕ Ω0 (Y ) ⊕ H 1 (Y ; R) . C(Y Anticipating the discussion from Chapter 3, we will fix a parameter λ ∈ R and include it in the monopole map as a perturbation of the first Seiberg-Witten equation..

(122) 2.3. Monopole map for closed 3-manifolds. 31. A preliminary version of the monopole map is thus given by ˜ ) → C(Y ˜ ), µ˜ λ : A(Y. (2.2) . . (A0,ψ , a, f ) 7→ (A0, DA0+iaψ + λψ , −i ∗FA0+ia − σ (ψ ) + d f , d ∗a + fh , ah ) (2.3) 1 7→ (A0, DA0ψ + λψ + ia ψ , − ∗iFA0 + iσ (ψ ) + ∗da + d f , d ∗a + fh , ah ), 2 where ah and fh denote the harmonic parts of a and f respectively. For simplicity, we will omit λ from the notation of the monopole map whenever possible. 2.3.2. Remark. A few words on the terms in the above definition. In its simplest form, the monopole map is defined using only the Seiberg-Witten equations1 , i.e. as a map Conn(L) × Γ(S ) → Γ(S ) ⊕ iΩ 1 (Y ), (A0,ψ ) 7→ (D A0ψ , ∗F A0 − σ (ψ )). After fixing a connection A on L we get an identification Conn(L)  Ω 1 (Y ) and the map translates to Ω 1 (Y ) × Γ(S ) → Γ(S ) ⊕ iΩ 1 (Y ), 1 (a,ψ ) 7→ (D Aψ + iaψ , ∗F A + ∗ida − σ (ψ )). 2 In order to interpret the monopole map as an element of some stable cohomotopy group, it is desirable to write it fibrewise as a sum of a linear Fredholm map and a compact map ([BF04, Theorem 2.6], [Bau04a, §2]). The linear part of the above version of the monopole map is given by (ψ , a) 7→ (D Aψ , ∗ida). This map has no chance of being Fredholm, because ∗d : Ω 1 (Y ) → Ω 1 (Y ) has infinite-dimensional kernel and cokernel. To remedy this, we include d : Ω 0 (Yf) → Ωg 1 (Y ) and its adjoint d ∗ : Ω 1 (Y ) → Ω 0 (Y ) in the definition. The resulting operator d∗d∗ d0 is elliptic (§1.6), and it has a well-defined index and a Fredholm extension to every Sobolev completion (§A.1.2). The other summands appearing in (2.2) and (2.3) influence the index (2.15) of the linear part l (2.12b) of the monopole map. Another important role of the additional summands (in particular, the projections onto harmonic forms) is to make the linear part on forms injective. Namely, the linear part on forms corresponds to the restriction of the monopole map onto the set of points fixed by the residual U (1)-action discussed shortly. If this restriction (i.e. the linear part on forms) is not injective, the restriction is not proper, and hence the monopole map cannot be proper (i.e. the desired boundedness property2 cannot hold). J 1 2. cf. [Sco05, p. 442] introduced on p. 38.

(123) Chapter 2. The monopole map on 3-manifolds. 32. ˜ ) and C(Y ˜ ) of the group G = {u : Y → U (1)} = There is an action on A(Y map(Y ; U (1)) of gauge transformations of L = det s which consists of the following actions: G × Γ(S ) 3 (u,ψ ) 7→ u · ψ ∈ Γ(S ),. (2.4a). S. u · A0 = A0 + 2udu −1 ,. A0 ∈ Conn(L),. (2.4b). u · ∇A0 = ∇A0 + udu −1 ,. A0 ∈ Conn(L).. (2.4c). The first action is given, and the others follow from the first. The action is trivial on forms. It is clear that the action of G on Conn(L) is not free, with stabilisers being the constant functions Y → U (1). The action becomes free if we restrict  to G0 B ker evy0 : G → U (1) . The monopole map µ˜ (2.3) is equivariant with respect to the action of the group of gauge transformations G, because this is true for the Seiberg-Witten equations. In particular, it is G0 -equivariant, and we get a map ˜ G0 : A (Y ) → C(Y ), µ = µ/. (2.5). which will be called the monopole map of the pair (Y , s). 2.3.1.1. Picard torus. Seiberg-Witten equations are invariant with respect to the action of G, so the solutions are considered up to gauge equivalence. In order to encode solutions only up to gauge transformations, we discuss the quotient of the space of all Hermitian connections on L with the same curvature by the action of the based gauge group. 2.3.3. Lemma. For a fixed Hermitian connection A ∈ Conn(L), the subspace A + i ker d ⊆ Conn(L) is invariant under the action of the based gauge group G0 . Furthermore, (A + i ker d )/G0  H 1 (Y ; R)/H 1 (Y ; Z)  Pics (Y ), where Pics (Y ) denotes the Picard manifold of Y . Proof. Since FA+ia = FA + ida, the subspace A + i ker d ⊆ Conn(L) consists precisely of those connections, which have the same curvature as A. Since the.

(124) 2.3. Monopole map for closed 3-manifolds. 33. curvature of a spinC -connection is invariant with respect to the G-action, the invariance follows. As mentioned earlier, the action of G0 on Conn(L) is free. In particular, it acts free on A + i ker d. Let (G0 )0 denote the connected component of the map u 0 ≡ 1. That is, (G0 )0 is the subgroup of G0 consisting of all maps u homotopic to u 0 (i.e. null-homotopic). As already mentioned, every u ∈ (G0 )0 can be written as u = e i f for some smooth function f : Y → R. Hence, u ∈ (G0 )0 acts on A by adding −id f , i.e. the (G0 )0 -orbit of A is of the form (G0 )0 · A = A + i im d. This implies   1 (A + i ker d ) ker d  (G0 )0 im d = H (Y ; R). On the other hand, there is a short exact sequence 0 −→ (G0 )0 ,−→ G0 −→ π0 (G0 ) −→ 0, u 7−→ [u]w , with [u]w ∈ π 0 (G0 ) = [Y , S 1 ] denoting the path-component of u ∈ G0 , i.e. its homotopy class. The fact π0 (G0 ) = [Y , S 1 ] = [Y , K (Z, 1)]  H 1 (Y ; Z),. (2.6). yields a short exact sequence 0 −→ (G0 )0 ,−→ G0 −→ H 1 (Y ; Z) −→ 0. In other words G0. . 1 (G0 )0  H (Y ; Z),. and finally (A + i ker d ). . . (A + i ker d ). G0 .  (G0 )0. . . G0. .  (G0 )0.  H (Y ; R) H 1 (Y ; Z)  Pics (Y ). 1. .  2.3.4. Remark. The space Pics (Y ) does not really depend on s (i.e. for every spinC structure we get a copy of the same torus). The notation only suggests that we have the copy corresponding to the chosen spinC structure. J.

(125) Chapter 2. The monopole map on 3-manifolds. 34. 2.3.1.2. Monopole bundles. The domain and codomain of µ are given by: ˜ )/G0 , A(Y ) B A(Y ˜ )/G0 . C(Y ) B C(Y Both A (Y ) and C(Y ) are infinite-dimensional vector bundles over Pics (Y ): π A : A(Y ) → Pics (Y ),   [A + ia0,ψ ], a, f 7→ [A + ia0],. . π C : C(Y ) → Pics (Y ),  [A + ia0,ψ ], b, д, ah 7→ [A + ia0].. (2.7a) (2.7b). (2.7c) (2.7d). The bundles A (Y ) → Pics (Y ) and C(Y ) → Pics (Y ) described in (2.7) are not trivial 1 in general (of course, if H 1 (Y ; R) = 0, then Pics (Y ) consists of a single point). 2.3.1.3. Monopole map on fibres. The monopole map µ in (2.5) is a fibre-preserving map between infinite-dimensional vector bundles over Pics (Y ) (because the induced map on the base space Pics (Y ) is the identity). However, it is not a vector bundle map because it is not linear on fibres. In the following, set A0 = A + ia0, with a0 ∈ ker d. The restriction   µ [A0] : π A [A0] → π C [A0] to fibres  π A [A0] =  π C [A0] =. (. ) ([A0,ψ ], a, f ) : ψ ∈ Γ(S ), a ∈ Ω1 (Y ), f ∈ Ω0 (Y ; R) ,. (. ) ([A0,ψ ], b, д, ah ) : ψ ∈ Γ(S ), b ∈ Ω1 (Y ), д ∈ Ω0 (Y ), ah ∈ H 1 (Y ; R) ,. is of the form3 :. 3. cf. [BF04, p. 11].

(126) 2.3. Monopole map for closed 3-manifolds. 35.   µ [A0] : [A0,ψ ], a, f   7→ [A0, DA0+iaψ + λψ ], − ∗iFA0+ia + iσ (ψ ) + d f , d ∗a + fh , ah =  0  1 1 [A , DAψ + ia ψ + ia0 ψ + λψ ], − ∗iFA + iσ (ψ ) + ∗da + d f , d ∗a + fh , ah . 2 2 (2.8) Note that the last equality holds due to a0 ∈ ker d, so da0 = 0. Also note that the fibres carry the obvious vector space structure: [A0,ψ 1 ] + τ [A0,ψ 2 ] B [A0,ψ 1 + τψ 2 ], τ ∈ C. Let ι H 0 denote the inclusion H 0 (Y ) ,→ Ω 1 (Y ). Over every point in Pics (Y ) (i.e. in each fibre) the monopole map µ can be written as the sum of the following assignments:   l : π A [A0] → π C [A0] ,     l : [A0,ψ ], a, f 7→ [A0, DAψ + λψ ], ∗da + d f , d ∗a + fh , ah ,   c : π A [A0] → π C [A0] ,     1 1 c : [A0,ψ ], a, f 7→ [A0, ia ψ + ia0 ψ ], − ∗iFA + iσ (ψ ), 0, 0, 0 . 2 2. (2.9a). (2.9b). The fibre of A(Y ) over [A0] ∈ Pics (Y ) = (A + i ker d )/G0 can be written as follows ( 0 ) 1 0 0  (A ,ψ , a, f ) : ψ ∈ Γ(S ), a ∈ Ω (Y ), f ∈ Ω (Y ; R) π A [A ] = G0  0 1 0  {A } × Γ(S ) /G0 × (Ω (Y ) ⊕ Ω (Y ; R)),  Γ(S ) ⊕ Ω1 (Y ) ⊕ Ω 0 (Y ; R),. (2.10). where the last isomorphism is given by [A0,ψ ] 7→ ψ with the obvious inverse. Note that we need to keep the representative A0 fixed in the definition of this isomorphism. The isomorphism clearly depends 2 on the choice of a representative of [A0]..

(127) Chapter 2. The monopole map on 3-manifolds. 36. Similarly, the fibre of C(Y ) over [A0] ∈ (A + i ker d )/G0 can be written in the following form  π C [A0] = ( 0 ) (A ,ψ , b, д, ah ) : ψ ∈ Γ(S ), b ∈ Ω 1 (Y ), д ∈ Ω0 (Y ), ah ∈ H 1 (Y ; R) G0   {A0 } × Γ(S ) /G0 × (Ω 1 (Y ) ⊕ Ω 0 (Y ) ⊕ H 1 (Y ; R)),  Γ(S ) ⊕ Ω1 (Y ) ⊕ Ω0 (Y ) ⊕ H 1 (Y ; R).. (2.11). In short, after fixing some representative of [A0] we get identifications  π A [A0]  Γ(S ) ⊕ Ω 1 (Y ) ⊕ Ω0 (Y ; R),  π C [A0]  Γ(S ) ⊕ Ω 1 (Y ) ⊕ Ω0 (Y ) ⊕ H 1 (Y ; R). Using these identifications, the assignments (2.8) and (2.9) now become maps Γ(S ) ⊕ Ω 1 (Y ) ⊕ Ω0 (Y ; R) → Γ(S ) ⊕ Ω 1 (Y ) ⊕ Ω0 (Y ) ⊕ H 1 (Y ; R), and are of the following form µ [A0] : (ψ , a, f )   1 1 7→ DAψ + ia ψ + ia0 ψ + λψ , − ∗iFA + iσ (ψ ) + ∗da + d f , d ∗a + fh , ah 2 2 (2.12a) l [A0] : (ψ , a, f ) 7→ (DAψ + λψ , ∗da + d f , d ∗a + fh , ah ), (2.12b) 1 1 (2.12c) c [A0] : (ψ , a, f ) 7→ ( ia ψ + ia0 ψ , − ∗iFA + iσ (ψ ), 0, 0, 0). 2 2 The notation µ [A0] , l [A0] , c [A0] reflects the dependence of expressions (2.12) on the point in [A0] ∈ Pics (Y ), as well as on the choice of the representative A0 = A+ia0 of this point. How do maps (2.12) vary with the change of a representative of [A0]? In other words, after applying the same procedure with a different representative for [A0], does one get the same maps Γ(S ) ⊕ Ω1 (Y ) ⊕ Ω 0 (Y ; R) → Γ(S ) ⊕ Ω 1 (Y ) ⊕ Ω 0 (Y ) ⊕ H 1 (Y ; R)? The answer is no. With the change of A0 = A + ia0, all maps (2.12) containing a0 change, i.e. µ [A0] and c [A0] change as A0 = A + ia0 varies. On the other hand, the presentation of the linear part l [A0] does not change 3 . In other words, the change of fibrewise presentation is solely detected by the non-linear part..

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